NASA Technical Reports Server (NTRS) 19760014913: Rotational excitation of symmetric top molecules by collisions with atoms: Close coupling, coupled states, and effective potential calculations for NH3-He

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cT <■

4*

ROTATIONAL EXCITATION OF SYMMETRIC TOP MOLECULES
BY COLLISIONS WITH ATOMS :

CLOSE COUPLING, COUPLED STATES, AND , EFFECTIVE POTENTIAL
CALCULATIONS FOR NH^-He.*

Sheldon Green

Department of Chemistry
Columbia University
New York, N. Y. 1002?

and

Goddard Institute for Space Studies

2880 Broadway

New York, N. Y. 10025

(£2 APR 1970 *3

r«v

,d>

RECEIVED

Na $A SR FACILITY

,NP W brahch

§?1

/

:5SJ

(NASA-TB-X-72994) ROTATIONAL EXCITATION OF N76-22001

SYBBETBIC TOP BOLECULES BY COLLISIONS WITH
ATOBS: CLOSE COUPLING, COUPLED STATES, AND

EFFECTIVE POTENTIAL CALCULATIONS FOB NH3-He Unclas

(NASA) 41 p HC $4.00 CSCL 20H G3/72 24782

Work supported by NASA Grant •!>. NSG 7105

ABSTRACT

The formal ism for describing rotational excitation in collisions
between symmetric top rigid rotors and spherical atoms is presented both
within the accurate quantum close coupling framework and also the coupled
states approximation of McGuire and Kouri and the effective potential
approximation of Rabitz. Calculations are reported for thermal energy
HH^-He collisions, treating as a rigid rotor and employing a unifonn
electron gas (Gordon -Kim) approximation for the intermolecular potential.
Coupled states is found to he in nearly quantitative agreement with
close coupling results while the effective potential method is found to
be at least qualitatively correct. Modifications necessary to treat
the inversion motion in are discussed briefly.

2

REPnODUCBIUTY OF Tttf!

original page is po»>

I , Introduc ticm

Recent advances in numerical techniques and computational capabili-
ties have made it possible to study energy transfer in molecular colli-
sions - at least for some simple systems - by obtaining essentially exact
numerical solutions to the quantum close coupling (CC) scattering equa-
tions for molecules interacting via realistic, e.g., ab initio , inter-
molecular potentials. Unfortunately, the expense of CC calculations
increases rapidly with the number of energetically accessible molecular
quantum levels, due, in large part, to the 2j+l degenerate sublevels
which must be included for each rotational level, Therefore most
calculations to date have been limited to the simplest case, collisions
of linear molecules with atoms, and to energies where only a handful of
the lowest molecular rotational levels are accessible.' 1 ' The hydrogen
molecule with its small moment of inertia has only a few rotational
levels below the threshold for vibrational excitation, and calculations

have been performed for vibrational- rotational excitation of Hg by eol-

2 '

lisions with atoms. Only one calculation for rotational excitation of
a linear molecule by another linear molecule (Hg-Hg) has been reported,* 0
and there has also been one calculation for rotational excitation of an
asymmetric top rotor by an atom (H^CO-He).^

Because much of the computational effort in CC calculations is due
to the degenerate rotational sublevels and because many collisional
phenomena are sensitive only to the average over these degeneracies, it
seems reasonable to sacrifice some information about these sublevels in
order to simplify the calculation. This notion is the basis for the

c £

effective potential (RP) approximation of Rabitz and the coupled states

3

(jC$ approximation of McGuire and Kouri. Both of these methods have now

been tested by comparison with available CC results for degeneracy-

averaged (i.e., state-to-state) integral cross sections, a(i-*f), in

5-7

linear molecule collisions. The CS approximation appears to be quite
reliable except, perhaps, when there is strong, long-range anisotropy
in the potential. Accuracy of the EP approximation appears to be more
sensitive to the strength of the anisotropy in the potential, being
better for weaker angle-dependence; however, considering that it is much
cheaper- than CS, it is probably still accurate enough to be useful for
mtny molecular systems. If these approximations can be shown to remain
accurate, they will be invaluable for understanding energy transfer in
other, more complicated systems where CC calculations will not be possible.

In this paper we consider scattering of a rigid symmetric top rotor
by a closed-shell, ^S atom. As an example, collisions of with lie
are considered, using a theoretical' elec troa gas approximation to the
interaction potential, and treating collision dynamics viUiin the accurate
CC framework as well as the CC and IIP approximations.

The ammonia molecule is a typical symmetric top, having a three-
fold axis of symmetry through the nitrogen. It is not, however, an
ideal example of a ri gld rotor because it undergoes rapid inversion -
a large amplitude vibration of the nitrogen through the plane of the
hydrogens. Tills inversion motion splits the normally degenerate sym-
metric top k-doublc*ts. Transitions between the inversion doublets are
readily observed at microwave frequencies making KII^ amenable to studies
of collisional energy transfer using the microwave double resonance
techniques of Oka,^ and a fair amount of rotational relaxation data is

therefore available for tills system.

For the purposes of this study the inversion doubling has been
ignored, and NH^ treated as a rigid symmetric top. Typical splittings
between inversion doublets are 20 - 25 GHz which corresponds, classically,
to a vibrational period of about 50 ns. Since the duration of a thermal
energy Nll^-He collision is 1 ns, the rigid rotor approximation may not
be unreasonable. Nonetheless, until this effect is considered in more
detail, results from these calculations should be viewed as modelistic,
and comparisons with the experimental data should be treated with caution.

Rather, this study was undertaken for the following reasons. There
has been no previous study of collisions between symmetric tops and
atoms, especially those systems dominated by short-range forces, where
the accuracy of the intcnnolecular potential and scattering approxima-
tions could be v/ell documented. Thus, despite a moderate amount of
double resonance data, energy transfer in such systems is still not well

understood, and "a quantitative theoretical treatment of the transition

8b

probability will be needed". For example, even the typical size and

shape of the short-range anisotropy are not known, (it should bo recalled

in this context that the anisotropy in linear jnoleculo-atom collisions

lc\

was recently found to be much larger than anticipated. ) Also ecsen-
tinlly \Uiknown are the relative magnitudes of different Aj and Ale tran-
sitions, and the cause of the observed parity "selection rules". As
indicated ubovo it is rapidly becoming possible to examine such questions
by accurate scattering calculations on ah ini t.l o intcrmolecular sui’faces.
Because of its low moment of inertia and hence widely spaced rotational

5

levels, and also because of ortho-para separation, it is feasible to
treat collision dynamics for NHy at least for the lower rotational
levels and thermal collision energies, within the accurate CC framework.
These calculations can then be used to test the accuracy of cheaper
approximations, such as CS and EP, which might then be used to study
collision dynamics for this and similar systems in more detail.

The CC, CS, and EP scattering formalisms for symmetric top-atom
collisions are reviewed in Section II since this does not appear to be
conveniently available in the literature. The numerical calculations
for NII^-He are presented in Section III. Section IV summarises the
major conclusions of these calculations and indicates briefly the
modifications necessary to treat the inversion motion.

6

II. Scattering Formalism

The total Hamiltonian for collisions of u rigid rotor and an atom,
in space-fixed coordinates located at the center of mass of the system
can he written as

II - H .(ft) + T(R) + V(Q, R) (la)

rot m **

where the kinetic energy of collision,

T(R) = -(A 2 /2u) c - l 2 ) , (lb)

can be separated into radial and orbital angular momentum contributions

as indicated. The rotor orientation Is specified by fMagy)* the

9

Euler angles that rotate space-fixed axes into the body-fixed, princi-
pal moment of inertia axes of the molecule; Die collision coordinate
from rotor center of mass to the atom is conveniently expressed in polar
coordinates as R=(R,0, $) ; u is Die reduced mass for the collision; and
ft i s Planck's constant divided by 2rr. The scattering wave function Is
obtained by expanding in the rotor eigenfunctions, which ure complete
in fi; spherical harmonics (partial waves) complete in (©,$); and radial
functions u(r). Substituting this expansion Into the Schrodinger equa-
tion leads to coupled second-order differential equations for u(r). The
usual CC method is obtained by transforming these equations to a total
angular momentum representation In Die space-fixed coordinates; the CS
and El’ approximations are obtained by ignoring or averaging over some of
the coupling terras . Details of these scattering formalisms will be
presented after discussing the rotor functions and the form of the inter-

r — » *« TTCTm ITV 0{1 TffF ,

P i- r<,„,

molecular potential.

A. Rotor vavofunc tlons.

The general rigid rotor Hamiltonian can be written

H rot ' + ( 21 2 + ( 21 3 ) ' 1 ^' 2 (2)

where 1^, I^, and 1^ arc moments of Inertia about the (molecule-fixed)

principal axes of inertia x', y’, and z 1 , respectively. (Primes will

be used to denote rotating, body- fixed coordinates.)

For a symmetric top molecule, two of the moments of inertia are

equal - I^=Ig, i.e., z' is chosen as the symmetry axis - so that Eq. ( 2 )

becomes

H st = ( 2 I i)"V + [( 2 X 3 ) " 1 -(2I 1 ) ” 1 ]^ z . 2 (3)

o p 2 2

where ~ 9. 1 » + P 1 is the total angular momentum of the rotor.

y y z

Eigenfunctions of 11^ can be labeled by J, k, and in, the total rotor

momentum and its projection on the body-fixed z'-axis and on the space-
fixed z-axis, respectively. Then

| Jkm> = j(j+l)fc 2 | Jkm> ,

(4)

| Jkm> = kft | Jkm> ,

(5)

and r J

a z

| Jkin> - nti | JIon> .

(6)

Comparing with Eq. (3) one finds that
”.t 1 ■ E jk 1 • 1tat>

(7)

vith

v ■ * 2

((2I 1 ) _1 j(j+l)H[(2l 3 )" 1 -(2I 1 )*' :i ]k ? }.

(8)

The symmetric top eigenfunc lions can be identified vith matrix elements
of the rotation operator,

| Jkm> = [(2J+])/(6 tt ? )] ' (9)

where, for historical reasons, we follow the convention of Tbaddeus^

.

8

that

( 10 )

RFJ’RODUCrEIT

ORIGINAL PAG

'nr or tiie

- is POOR

J>jM - e 1 - d J( P )

1 9

with as defined by Edmonds. The total angular momentum quantum

number J may take any non-negative integer value; k and m, being projec-
tions of J, are restricted to positive or negative integers whose abso-
lute value is less than or equal to j.

For future reference, the only non-zero matrix elements of the
general (asymmetric top) rigid rotor Hamiltonian, Eq. ( 2 ), in this basis
are

<Jlan | H rot | Jkm> = / I 2 {[(2I 1 )" 1 +(2l 2 )" 1 ][j(j+l)-k 2 ]/2+(2I 3 )" :i k 2 ) (llu)

and

<J k+2 m | H ro ^ | Jkm> - <Jkm | | J k+2 m> =

(ft 2 A)[(2I 1 )’ 1 -(2I 2 )' 1 ]{[j(j+l)-k(k+l)][j(j+l)-(k+l)(k4B)])^ . (lib)
For the symmetric top, it can be seen from Eq. (8) that | Jkm> and
| J-kra> arc degenerate no that any linear combination will also be n
valid eigenfunction with the some energy. Proper symmetric top wave-
functions must also be eigenfunctions of the inversion operator, and the

correct linear combinations are

1

| Jkme> » [2(H6 k0 )]" ,? ( | Jlrm> + c | J-)on>) (l2)

where now k^O, and e-jd except for k r -0 when oiily e“+l is allowed.

Decuuse the | Jkm> are a complete set in the angles of orientation,
ft, wavefunctions for the general, asymmetric top can also be expanded
in this set. It is found, c.f. Eq. (ll), that the asymmetric top
Hamiltonian will not mix states with different J,m but only different
vulucc of k. Tims, asymmetric . top wavefunctions can be written as

| Jm T > = T y a | jkra> . (l3)

9

I

It is convenient to develop the scattering fomaliGm for "primitive"
symmetric top functions, | Jkm>. Because k pla.ys the role of a "specta-
tor" in the angular momentum coupling, and because the Schrodinger equa-
tion is linear, it will then be straightforward to take the appropriate
combinations, Eq. ( 12) or Eq. (l3), to obtain the correct formulation
for symmetric top or asymmetric top rotors. CC, CS, and EP scattering
formalisms will be developed in Sections IIC, IID, and IIE, respectively.
Correct symmetri nation for symmetric tops will be considered in more

det '1 in Section IIE. Extension to asymmetric tops is straightforward

4

and las been discussed by Garrison.

B. Tn ter molcculn r potential .

The interaction potential, V(f), R) in Eq. (l) is a function only of
the relative orientation of the projectile with respect to the rotor.

Thus, V(f),R) « VCR') c VCR',©',*') where R' is the position of the pro-

A/ ^

jectile with respect to (body-fixed) axes along the principal moments
of inertia of the rotor. It is convenient to expand the angular depen-
dence here in spherical harmonics,

V(R' ,0' , $') « E v (R') Y, (0’,*') . (l4)

Xu *u

One can then use tin; transformation properlies of the spherical harmonics

9

to relate body-fixed and space -fixed coordinates.

V(n,R) - V(K')

A' A/

( 1 '))

\ V n,) rjk>

Xu(R) ,

where the fr ct that R-R' has been used.

10

The phace convention for spherical harmonics,

Vu C. * (E) - ( - )U V R)# • <16)

IjlUS the fact that the interaction potential is real imply that

v. (R) - (-) U v (R)* . ( 17 )

a>-u XM

For most, systems it v/ill also be possible to choose the principal moment
of inertia axes so that the v (R) are real. This requires only that
the x'z'-plune be a (reflection) plane of symmetry so that

V(R',0',$') « V(R' ,0' ,-b') (18)

which will always be possible for molecules that are symmetric tops due
to an n-fold axis of symmetry alone z'. The interaction potential can
then be written as

v(R\0’,*') - v^ (R1) t'y )w (R')^(-)% ; _ u (R')] (19)

* 'V 1

V* ^ (2 'V r Xu(coa9') cor.y}' ,

with P, an associated Legendre polynomial.

It is important 1 o emphasise that the body-fixed coordinaie system
used to expand the jxotential must correspond to the principal moment of
inertia axes (x'jy'^') used to define the rotor wave functions.

C. Close Coupling .

The development of the close coupling equations for symmetric tep-

atom collisions follows the development for linear molecule -atoms of
Arthurs and Palgarno^' 1 very closely, so that it will only be outlined
below. One can form coupled, total angular momentum functions

11

RKPRODUCW TV OF TBS

ORIGINAL IV* ’ ’3 P<X)B

| | JM> | Jkm> | (?0)

where | Jkm> are gi ven by Eq. (9)> the "partial wave" functions are

( 21 )

9

«V ■ V ( ®’ S) >

V

and ^^WiJgTJig | is a Clcbsch-Gordan vector-coupling coefficient.

The scattering vuvcfunction with total momentum J and Projection M
on the space-fixed z-axis, and appropriate to the entrance channel Jk£

satisfies a Schrodinger equation

[ H st

W M h + v - E - V

JM

* 0

( 22 )

where E is the relative kinetic energy and E^ is given by Eq. (0).

Expanding Y

JM

JH

JM

‘JU

R' 1 u, |

(23)

and substituting into the Scnrodingcr equation gives the usual coupled
equotiono for the radial functions:

r <£. 4 . 2 T!HVf

C ffl 2 ‘ R 2 J'k'Jk - 1 Vk-f (E) "

(ty/t?) E .y,^, <JMj"k"-t" | V| JMJ'k'-t'> (2*0

where the wavenumber is given by

?

- (aiA") (K+E..-E.,. ,) ,

(?^)

J 'k' Jk 7 '" “Jk "J 1 k 1

and the coupling matrix element Is

Vxy (R,) V r<,) I \ n (H > (26)

Xu

(l V \\ /J J' XN O’ J-)

0 Oj (k -k’ u] \l j X J’

with (:::) a j’» - J symbol und {:::] a 6-J symbol.' There is no coupling
between different JM, and tlie matrix element in independent of M which

3 ?

will subsequently be dropped from the notution. It in readily nhovn

that

(2V)

<JJki| Y^ | JJ'k’t'> * | (-) M Y )o .^| J d k O

which, together with Eq. (17) ensures that the couplinc matrix ic
Hcnnitcon. Kote that different vuluer. of k are connected only by terms
in the potential ouch that (j^k'-k. The following property of the
coupling matrix element 8 will also bo useful:

<JJ-U|Y X | JJ ' -k 1 1'> - (-) j4j,+X <JJktl Y x _^| (?G)

Comparing the asymptotic form of u(l») with the intcructionlcs::
solutions defines the: scattering S-matrix in the usual v/uy:

u. , # , JJU '(]<) ~ 6^,6^,*, ex P r-i(x JkJk R-^t/2)]

'JJ' kk’^U’

(?9)

-(><.. .. /k iM , ».)=<jkt| S J I j'k't^ t^p[i(x , , .K-t'n/") j.
State-to-stute integral cross sections, summed over final and averaged
over initial degeneracies, can be obtained from the E-matrix an

o(jK-j'k') " E J U ’

Cio)

It should be noted that the k quantum number pluyr a "spectator"
role in the angular momentum coupling. Therefore, one can immediately
write do\/n fosiulao, in terms of E-matrix elements, for di fferent ini cross
sections, prcsr.uj broadening, etc., by me rely adding u 1; label to
the foix.ulus for linear molecules. Furthermore, if the rotor wave-func-
tion:; tut* linear combinations among k values, as ic the care for symmetric
top functions of proper symmetry and also for asymmetric tops, il in
necessary only l<> tronsfoiw the coupling matrix, r.q. (fO, into this now
basis and replace the k label with the appropriate new label (’.•■• kc f >r
symmetric tops, c.f. Eq. (if) ; or t for trie toj ;, c.f. Eq. (13)).

13

D. Coupled States .

It is possible to write the CC scattering equations in a body-fixed,

i.e., rotating, coordinate system rather than in the space-fixed system

of Arthurs and Dalgarno. (Sec, e.g., the helicity formulation of Jacob
12 \

and Wick. ) The CS approximation is derived in the "R-helicity"

coordinates in which the z’-axis lies along the collision coordinate R.

Then m , the projection of the orbital angular momentum on R, is zero
•L /v

by definition so that the total angular momentum expansion basis, analo-
gous to Eq. (20). can be labeled by | JMjkno where m is the projection
of j on R. This basis is related to the space-fixed basis, Eq. (20), by

/V

13

a unitary transformation, as discussed, e.g., by Child ~ and Walker and
l4

Light. In the body-fixed system, the interaction potential matrix is

— 2
diagonal in m and independent of However, £ is no longer diagonal

as it is in the space-fixed system, but connects states with different

m, corresponding to "Coriolis forces" in the rotating frame.

6 a

The coupled states approximation introduced by McGuire and Kouri

2

consists of neglecting the off-diagonal matrix elements of {. in the
body-fixed frame and further approximating the diagonal ones by

<JMjkm | £ | JMj’k'm^ - * ? j(j+l)5~, 6 , . , 6, , . (3l)

nim J J Ki\.

Tins approximation and some related variants have been extensively dis-
cussed by McGuire and Kouri and co-workers.^ Therefore, only the modi-
fications to the potential matrix elements necessary for symmetric tops
vi.ll be preserved here. The potential matrix in the body- fixed frame
is given by

lh

I

OMJkml Ei v (R«) Y.(R') I JMJ'k'mV *= E v (R ) (-)" k '" ID (32)

1 Xu Xu Xu 1 x Xu Xu

which is readily derived using the fact that

Y X|1 (3y) = (-) u [(2X+l)/(4n)]2 ^ 0 x (apy). (33)

Note that Eq. ( 32 ) implies no coupling unless m=m', and also that dif-
ferent values of k are coupled only by tems in the potential such that
jj=k-k'. Again, matrix elements are independent of M <i ich is subsequently
dropped from the notation.

As in the CO method, the potential matrix is Hennitean since,

c.f. Eq. (27),

| Y^ | Jj'k’mV = <Jj 'k'm' | (-)^ | Jjkm> . (34)

The follov.'ing properties of the potential matrix elements will also be
useful:

<J^k-m | Y^| Jj 'k' -m'> = (-) J+J NX <Jjk^ | | JJ'k'SV , (35)

and

<Jj-km|Y | Jj'-k'm^ = (-) J+J ' +X <iJJki I Y I Jj'k'iV . ( 36 )

XU X j ~u

The coupled differential equations for the CS approximation are

identical in form to those for CC (c.f. Eq. (24)) except that t=J and

the channels are .labeled by Jjkm rather than Jjkj,. There is no coupling

between channels with different in, and separate sets of equations must

be solved for m=0, +1, +2, . . +J*, where j* is the maximum rotor j-value

of interest. The asymptotic form of the radial functions defines a

scattering i.atrix <jkm | S lT | j'k'mV exactly as in the CC method (c.f.

Eq. (29)) from which the degeneracy averaged cross sections are obtained

[ 2 . 1 + 1 ) (2.1 1 +l) (2X+l)

( J X y \ ( J x y \

\-k u k'y v-m 0 mV

i x y

15

%r

(c.f. Eq. (30)) as

o(jk-j'k’) - TT H JkJk " 2 (2J+l)” 1 Sjs (2J+1)

I 6 l 1j' 6 kk' " I ^ I J ' k ’“ > I 2

(37)

E. Effective potential.

5 &

The effective potential approximation was obtained by Rabitz by
taking a particular average of the potential matrix elements over degen-
erate rotational sublevels in an uncoupled, space-fixed representation.
This approach can alternatively be considered to be an expansion of the
total scattering .wavef unction with "effective rotational states" which
are non-degenerate and which do not couple to the (partial wave) orbital
angular momentum. This leads to coupled equations identical in fonn to
CC (c.f. Eq. (24)) except that, since J and g are now identical, the
channels can be labeled by Jjk rather than Jjk£.

The ET matrix elements for symmetric top systems have been derived
by Tarr and Rabitz. For a symmetric top-atom collision the potential
matrix is given by

1 v (R,) V R,) 1 JJ,k,> = \ V R } ( - )nJ ' k (38)

exp[in( | J’-j | +J’+j)/ 2] (4n)‘ a '[(2J+l)(2J , +l) ] 4

( i * JS
V-k 11 kV .

The potential r.iatrix is seen to be independent of J. Coupling between
k levels is due to terms in the potential such that ir-k’-k. Hermiticity
of the coupling matrix is ensured since, as in CC and C3,

<JJ'k’ | Y Xq J Jjk> = <Jjk | (-) W Y x _ m j Jj'k'> .

lf>

(39)

r

REPRODUCIBILITY OF THE
ORIGINAL PAGE IS POOR

The following property will also he useful:

<JJ-k | Y^| JJ»-k'> = (-) J+J ' +X <iJJk| Y x | JJ 'k'>. (4o)

The asymptotic foim of the radial equations defines a scattering
matrix <Jk | S J | J'k'> exactly as in CC (c.f. Eq. (29)) from which degen-
eracy-averaged cross sections are obtained as (c.f. Eq. (30))

o(jk-J , k') = i') Ej (2J+l) (4l)

I -<JH s ' r l J’ k '>l 2 •

The "covinting-of-states" correction, g(j,j')> was introduced by Zarur
5c

and Rabitz to ensure proper detailed balance,

(2j+l) h. 2 o(j-j') - (2j’+l) Ky 2 o(j'-j) , (^2)

which doesn't arise naturally in EP due to the vise of "effective rota-
tional states" that lack the normal 2J+1 degeneracy. Zarur and Rabi tz
suggested for linear molecule-atom collisions the most symmetrical form,

g(j,j') = C(2j+l)/(2j , +l)f . (43)

7

However, it appears that a better choice might be

1 >

(2^l)/(2J'+l),j>j' .

(44)

In either case, the counting-of- states factor is the same for symmetric
tops and linear molecules.

F. Symmetric top functions of proper parity .

As indicated in Section IIA, symmetric top eigenfunctions with
proper parity are, Eq. (l2),

| jkme> «= [2(1+6^)]"^ ( | jkm> +e | J-km> )

\

where k-'O and e-+l except that only e~-il is allowed for k=0. By consi-

17

dering this linear combination along with the symmetry properties of the
interaction potential, Eq. ( 19) , the final working equations are derived
for CC, CS, and EP scattering of a symmetric top molecule by an atom.

In the CC method, one must consider matrix elements of the inter-
action potential of the form

v \u I [ V ( - )M V U ]/( 1 V ' = (1 ' 5)

% [2(1 V rl[(1+ V (1+ W) 4 '

4€'<JJU| Y +(-) |1 Y x I JJ'-k'f>4ee'<T4-U| |

Note that n, k, and k' are non-negative here. Recalling that a matrix
element vanishes unless n=k'-k and using Eq. ( 28 ) to reverse the sign of
k and k 1 in the third and fourth terras, one obtains

% <^*1 cv ( - )Uy x-uV (1+ V) l w

(uxJJUl Y | Jj'k^’>+e<Ij-kt | Y^ | Jj’k't'>)

where

9 - [l+ee'(-) J+J,+X 4 U ]/ 2 . (47)

There will be a contribution from the first term in Eq. 0t6) only if
+|i=k' -k, and it will enter with parity

1,

k'-kiO

(48)

0) - <

{ (-) U , k'-k<0 .

There will be a contribution from the second terra only if ji-k'+k. Only
one of these terras will contribute unless k-0 and/or k'-O.

It can be verified that the potential matrix elements vanish iden-

tically unless

('*9)

The CC equations can therefore be split into two non-interacting parity
blocks for each J, leading to a reduction in computational effort.

The analysis for the EP approximation follows identically that for
CC. The final equation is

<JJke | t* Xu + (-)\^ J ]/(l + » 1J p) I JJ’k'O ■= ( 50 )

(uKJjk | Y | | Y^ | JJ'k’>)

with <? and cu as defined by Eqs. (47) and (46).

• The analysis for the CS approximation is similar except that the
role of k and k’ is interchanged:

<Jjkms | [Y XM +(-)hf x _ M ]/(l+6 ii0 ) | = (5l)

^ ^ 1+6 k0^ 1+6 k'0^ *

(a5<Tjkm | ’| Jj 'k'n^+e *<JJkm | Y^ | Jj'-k’m^) .

where

f 1, k-k'aO (52)

u) - 4

( (-) u , k-k'<0 .

and <? is given by Eq. (4y).

19

reproducibility of the
ORIGINAL PAGE IS POOR

III. Calculations for NH^-IIe
A. Intermoleculcr potential .

The intermolccular potential was obtained ns a function of the
position of He relative to the ammonia center of mass, in

body-fixed, principal moment of inertia corrdinates (c.f. Eq. (l4)).

The geometry of the ammoniu molecule can be described by the (carte-
sian) coordinates of the nuclei in this same system - N(0., 0., 0.127))
IL^ 1.771, 0., -0.5928), H 2 ( -0.8855, 1.5337, -0.5928), H 3 (-0.8855,
-1.5337, -0.5928) with all distances in bohr (l a Q = O.529l0xlO ^ cm) -
which is close to the equilibrium structure inferred from microwave

spectra. The interaction potential was computed via the electron gas

15

model of Gordon and Kim, using the computer progrnn of Green and

Gordon. ^ From comparisons with more rigorous quantum calculations for

other systems it appears that this method provides a semi -quantitative

estimate of the distance and angle dependence of the short-range repul-

17

sive interaction. It also appears to give a reasonable estimate of
the position of the van dor Waals minimum. It does not, however, cor-
rectly reproduce the long-range induction and dispersion energies, but
falls to zero too rapidly. 3u previous work we have supplemented the
electron gas approximation by smoothly joining it to the asymptotically
correct long-range electrostatic interaction. Because the cross sec-
tions here are not expected to be very sensitive to the long-range
interaction, and because thin study is designed more ns a model calcu-
lation than an attempt to obtain detailed cross sections for this
system, the electron gas interaction has been used without further

modification.

The electron charge densities necessary for the Gordon-Kira method
were obtained from Hartree-Fock functions. The NII^ function vac obtained
using a Gaussian basis of approximately "triple zeta" quality plus p
polarization functions on the hydrogens."^ For He, the accurate Slater

19

basis function of Clement! was used. Hie interaction potential was
computed for R'/a 0 «= 3.0 (0.5) 9.5, ©' = 0 (30) 100°, and $’ = 0 (15) 60°.
For each R' value, the angular dependence was expanded in spherical
harmonics. Fa. (19), by minimizing the root-mean- square average error.
Because of the three-fold axis of symmetry, only V with |l-3n are
allowed in the expansion. A twelve term fit, which reproduced all
computed points to a few percent, vac adopted for the scattering cal-
culations; this is presented in table .1 . Smooth radial functions and
derivatives were obtained from the tabulated points by fifth-order

Lagrange interpolation plus exponential extrapol.ation at short-range

lc

and inverse power extrapolation at long-range.

By examining semi -logarithmic plots of the v (R 1 ) for the chort-

Aji

range repulsive interaction, i.e. one finds approximate expo-

nential behavior

v. tv a. exp( -b. R') .

>41 X|J AU

(53)

Furthermore, the slopes of the dominant teims arc quite similar so that

(5*0

The strength of the anisotropy is ihen reflected in the size of the

V XM' a °)4i U 00 exI>( - b 00 n,) •

various a, , from which it is seen that the interaction here is only

Xu

weakly anisotropic, i.e., a «1.

B. Scattering calculations .

Using the intermolecular potential described in the previous sec-
tion CC, CS, and EP calculations have been performed. The coupled
differential equations were solved using the piecewise analytic algo-
rithm of Gordon 2 ^ as implemented in the MOLSCAT 21 computer programs.
Tolerances were set to obtain 2-3 significant figure accuracy in the
final cross sections.

Because of permutational symmetry among the identical hydrogen
nuclei in ammonia, the rotationul levels can be divided into two sets
which are associated with different nuclear spin states and which inter-
convert at a negligible rate in thermal energy collisions. levels with
k=3n are designated ortho-NH^ and levels with k-3n+l are designated
para-IlHg. That the scattering calculations predict no interconversion
of ortho- and para-NII^ can be seen as follows: Molecular symmetry ensures
that the expansion of the intemolecular potential, Eq. ( 19), will con-
tain only terms with pi“3n. The coupling matrix element s in the scattering
formalism vanish unless Ak=ji. Therefore, there will never be coupling
between ortho- and para-levels and hence zero probability for collisional
intercon version. Since ox tho- and par« -1'TII arc totally decoupled, scat-
tering calculations can be done separately for tlje two species. (This
situation is entirely analogous to the more familiar ortho-para distinc-
tion in lip.)

For pnrn-NU^ the scattering equations are invariant to simultaneous
change of parity in the incoming and outgoing channels. Thus o( jkc-*J 'k’e ' )
is equal to o( 'k'c ' ) where c denotes the opposite parity from c,
and only one of these cross sections will be reported subsequently, hole,

22

nonetheless, that both parities must be included in the scattering
calculations. This symmetry does not obtain for ortho-NH^ because the
k=0 levels exist in only one parity. Also, any energy splitting of
the k-do\iblets, such as that caused by inversion motion, vould destroy
the perfect symmetry found here for para-UH^.

The major approximation in the CC method is retention of only a
finite subset of the molecular rotational levels in the expansion of
the total wave function. Therefore, a number of calculations have been
done to examine convergence of the cross sections as the expansion basis
is increased. Rcsxilts of such tests are presented in table 3 for orlho-
NII^ and in table 4 for para-THI^. The difficulty in using larger basis
sets can be appreciated by noting that the 1315 calculation for ortho-
required solving 6l and 68 coupled equations for the tv:o parities
at each J, end the B20 calculation for para-NH , required solving two
sets of VC coupled equations for each J. It is apparent that converged
cross sections are obtained hy including all open (i.c. energetically
accessible) channels plus, perhaps, a few of the lowest closed channels.
Tlris rapid convergence can he attributed in part to the small anisotropy
and widely-spaced energy levels in this system which minimize the effect
of indirect (higher order) coupling. If the- interaction were more aniso-
tropic, or if the potential well were deeper compared to the energy
splittings (leading to Fcshbach resonances), one might expect poorer
basis set convergence. The basis set convergence in CS and RP calcu-
lations wan found to be similar to that presented for the CC calculations.

23

An examination of the CC cross ccctionG begins to reveal interesting

"propensity" rules for different AJ, Ak, and parity transitions, the

existence of which had been previously inferred from double resonance

data. To systematically study these, however, requires cross sections

among more levels than have been included here, so that apparent trends

have some statistical validity. Since it is not currently feasible to

extend the CC calculations to many more .levels, it is important to

document the validity of cheaper approximations. Toward this end, CS

and EP cross sections arc compared with CC values in tables 5 and 6.

The EP cross sections are given both with (Eq. 0*3)) and without (Eq.

(Ml)) the "counting- of-ctates" correction suggested by Zarur and
5c

Rabitz. The CS approximation is seen to give integral cross sections
in sciwi- quantitative agreement with CC. Although significant differences
exist for a few isolated transitions, CS appears to be generally quite
reliable for predicting the magnitude and relative size of different
cross sections for this system. The cheaper ET approximation is in
Gomewhat poorer quantitative agreement with CC but is still generally
reliuble for estimating the magnitude and relative size of cross sections.
Experience with linear molecule -atom collisions would lead one to attri-
bute the accuracy of EP here to the small anisotropy in thi s system.

Also, following previous experience, for most transitions EP without
the "counting-of- states" correction gives better ugreement with CC.

The integrjJ. cross sections considered ubove arc only one measure
of the overall collision dynamics - albeit a very important one since
they determine the rate of colliolonnl energy transfer - and it is

important to determine whether CS and EP provide a reliable description

• If ??

of other details of the collision dynamics us well. * In this paper

only the partial opacities - the contribution to the cross section from
different total J or partial waves - will bo considered. j.t is found
that the CS partial opacities agree very well with CC values for inelas -
tic processes, tlie largest discrepancies, if any, being in the lowest
few partial waves. For clastic cross sections, however, the agreement
is not so good, with the exception of o^(00+-»00 +) ; for other elastic
transitions CS values tend to oscillate about the CC results. Fig. 1
shows typical examples of both cases. The behavior of KP partial

opacities was found to be quite similar to that of CS. Although detailed

22

calculations have not yet been performed, one can anticipate from the
behavior of the CS and KP clastic partial opacities that these approx-
imations may not be adequate to describe other colli cicnol phenomena,
in particular, differential cross sec ms and collision induced spectral
pressure broadening, for this system.

IV. Conclusions

The formulism for describing collisions of a symmetric top rotor
with a spherical atom has been presented both within the accurate close
coupling framework and also the coupled states and effective potential
approximations. Calculations have been performed f' a model of the
NHg-He system which demonstrate the feasibility of CC when only a few
of the lowest rotational levels aia of interest. These begin to show
interesting "propensity rules" for parity changes, ihe existence of
which had been previously infer *ed from microwave double recononco
experiments.^ Unfortunate] y, a. proper study to compare with available
data requires more rotational level c thau can be bundled by CC with
present computational techniques.

On the other hand, it war. found that the CS approximation gave
results in nearly quantitative agreement with CC and that the KP approx-
imation v/nn at least qualitatively correct. Those methods are also
cheap enough to allow p proper study of collision'll energy transfer in
ammonia. To compare with experimental data, however, such a study must
also account for the inversion motion which has been ignored here. To
conclude this paper a brief discut. sion v/111 be given of ihe necessary
modi fi cut J o)is.

23

To a first approximation, inversion can bo described as one-
dimensional motion of the nitrogen nucleus along the yd -cals (with
cor res riding motion of the hydrogens in Ihe opposite direction to
preserve the center of mars at the origin). If this coordinate is
labeled h, then tlic rigid ro t6r configuration described in Section ITT. A

reproducibility of the

ORIGINAL PAGE IS POOR

corresponds to h*h e > the equilibrium position. From symmetry, the
vibrational potential energy is unchanged on taking h -• -li so that
there is n corresponding minimum at h*-h t , and a potential barrier between
those with maximum at h-0. The lowest, vibrations] mode in this poten-
tial in symmetric with respect to h -h, and the first excited mode
is anti- symmetric. To achieve proper symmetiy of Die tot a] wave fun ca-
tions for tsnmonin ( elec tronic-vibrational-rotutional -nuclear spin) the
symmetric (anti -symmetric) inversion function can combine only with one
of the ‘two | Jk+cm> symmetric top functions; the lower (upper) inversion
doublet then corresponds to the lowest (first excited) vibrational mode
plus the appropriate rotutional purity function. For k~0 only one of
the rotational parity st- ten exists und only the corresponding even or
odd vibrational state is then allowed. (See Ref. 23 for a more detr iled
discussion.) It is thus seen that each of the parity rotor basis func-
tions used in this study corresponds to a specific upper or lower member
of a doublet.

..

It is also necessary to consider tne effect of the inversion motion
on the interaction potential and Its matrix elements. For the projectile
at (R 1 ,0',*' ) in the molecule-fixed axes it i: apparent that the inter-
action depends parametrically on the inversion coordinate, h, tlje p >ten-
tial given in Section 111. A corresponding to h r h . From simple geometry

C 4

considerations one has the useful fact that the coordinate change h -• -h
Is equivalent to 0* — n< ’ . Th< potential t ' •' ■ elements must be ver-
aged over th- inversion motion. In principle, the interaction must be
oftained tts a functi in uf li 1 , 6 1 , and h and averaged over the vibra-
tional functions which depend on h. It can do noted, h ivcvt r, th t tl

lowest and first excited vibrational functions for ammonia peak rather
sharply near the two equilibrium positions h=+h e . To a good approxi-
mation then, it is seen that vibrational matrix elements between levels
with the same parj ty are given by

< + | V | + > » |[v(R , 0 , f t ;h )+V(R' ©*$‘;-h )] (55)

« i[V(R*0’ $’ jh^+vCR'jTT-®^' ;h e ) ] ,
and between levels of different parity by

<*| v| + >« Kv(R , 0 , $ , ;h e )-v(R , , n -0 , ,$ , ;h e )] . (56)

Thus, the computational techniques now appear to be in hand for

studying energy transfer in ammonia- rare gas collisions. The experi-

8

mental results of Oka will then provide a stringent test of these
theo ret leal methods. Besia.es the double resonance data, colli sional
excitation of ammonia is also of current astrophysical interest for
interpreting the observed non-thennal excitation of interstellar Nit,. '

For these reasons we hope to undertake a more complete study of this
system in the near future.

28

Acknowledgments

The early part of this work was done while the author was a
National Research Council Resident Research Associate supported by
NASA. I am indebted to Patrick Thaddeus for his continuing interest
in and support of this research and for insights into Euler angles
and rotation matrix conventions; to Richard N. Zare for an illuminating
discussion of parity in symmetric tops; to Barbara J. Garrison for
providing a copy of her thesis; to Sue Tarr and Herschel Rabitz for
communicating their work on the effective potential approximation; and
to William A. Klein for many stimulating and informative conversations
while this manuscript was being prepared.

?9

REFERENCES

1. a) R. Shafer and R. G. Gordon, J. Chera. Phys. 58, 5422 (1973);

*V\J

b) S. Green, Physics 76, 609 (1974); c) S. Green and P. Thaddeus,

A/V

Aotropbys . J. in press; d) S. Green and P. Thaddeus, Astrophys. J.
191, 653 (1974); e) S. Green, Astrophys. J. 201, 366 ( 1975) ; f) S.

/vw*'

Green and L. Monchick, J. Chem. Phys. ^63, 4198 ( 19 79) J c) B. T.

Pack, J. Chem. Phys. 62, 3143 (1975).

2. a) W. Eastes and D. Secrest, J. Chem. Phys. 56, 640 (1972);

b) J. Schaefer and V/. A. Lester, J. Chem. Phys fe, 1913 ( 19 75 )
and references therein.

3. S. Green, J. Chem. Phys. 62, 2271 ( 19 75 ) •

4. B. J. Garrison, W. A. Iester, W. H. Miller, and S. Green, Astrophys.
J. (Letters) 200, L175 0975); B. J. Garrison, Fh. D. Thesis, Univ.
of California, Berkeley, 1975, unpublished.

5. a) U. Rabitz, J. Chem. Phys. 57, 17l8 (1972); b) G. Zarur and II.

/WV

Rabitz, J. Chem. Phys. 59, 9^3 (l973)j'c) G. Zarur and II. Rabitz,

/VW

J . Chem. Pliys. 60 , 2097 (1974); d) S. Tarr and H. Rabitz, private

AW*

communication, 1975.

6. a) P. McGuire and D. J. Kouri, J. Chem. Phys. 60, ?488 (1974);

b) P. McGuire, Chem. Thyc. Letters 23, 575 (.1973); c) 1>. J. Kouri,

and P. McGuire, Chera. Phys. Letters 29, 4l4 (1974); d) P. McGuire,

"Validity of the Coupled States Approximation for Molecular

Collisions", preprint ; e) D. J. Kouri, T. G. Heil, and Y. Shiiuoni,

"On Refinements of the J r -conserving Coupled States Approximation",

z

prepri nt .

30

7.

a) S. Green, J. Chem. Fhys. 62, 3568 (1973), b) S. Green, Chera.

Phys. Letters, in press; c) S.-I Chu and A. Dalgarno, J. Chem.

Phys. 63, 2115 (1975).

8 . a) T. Oka, Adv. At. and Mol. riiys. < 9, 127 ( 3.973) ; l>) T. Oka, J.

Chem. Phys. 49, 3135 (1968).

AVv

9. A. P. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, N.J. :
Princeton Univ. Press, i 960 ).

10. P. Thaddeus, Astrophys. J. 173, 317 (1972).

11. A. M. Arthurs and A. Dalgarno, Proc. Roy. Soc. (London) A 256 , 5^0

• (i960).

12. M. Jacob and G. C. Wick, Ann. Phys. 7, 404 ( 1959) -

13. M. S. Child, M olecular Col lision Th eory (London: Academic Press,

1974) .

14. R. B. Walker and J. C. Light, Chem. Phys. J, 84 (l975).

15. R. G. Gordon and Y. S. Kim, J. Chem. Phys. 56 , 3122 (1972).

1 6 . S. Green and R. G. Gordon, POTLSURF, Quantum Chemistry Program
Exchange, University of Indiana, Bloomington, Indiana, program 251.

17 . S. Green. B. J. Garrison, and W. A. Lester, J. Chem. Phys. 63 , 13-54
(1975); G. A. Parker, R. L. Snow, and R. T. Pack, Chem. Phys. Letters

33, 399 (1975); Coe also Ref. le.

18. The wuvef unction was provided by H. Schor (private communication)
using a basis set adapted from R. G. Body, D. S. McClure, and E.
Clement!, J. Chem. Phys. 49, 4916 ( 3 . 968 ).

19. E. Clement i, IBM J. Res. and Develop. 9, 2 ( 1965 ).

31

20. R. G. Gordon, J. ohera. Phys. 51, lU ( 1969 ); R. G. Gordon, Methods
in Comput. Phy3. 10, 8l (l97l)> Quantum Chemistry Program Exchange,

AV>

University of Indiana, Bloomington, Indiana, program 187.

21. the M0I£CAT programs are designed to facilitate coupled channel
scattering calculations for molecular collision dynamics. Much of
the original computer code of R. G. Gordon (c.f. Ref. 20) has been
incorporated for solving the coupled differential equations.

22. L. Monchick and S. Green, "Validity of Approximate Methods in
Molecular Scattering. III. Effective Potential and Coupled States
Approximations for Differential, Gas Kinetic, and Pressure Broadening
Cross Sections", in preparation.

23. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (New York:
McGraw-Hill, 1995).

2b. M. Morris, B. Zuckerman, P. Palmer, and B. E. Turner, Astrophys.

J. 186 , 501 (1973).

/VvV

32

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Table 2

Rotational levels of ortho- and para- ammonia included in the

scattering calculations,
ortho

level

1

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2

10 +

3

20 +

4

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5

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19.8305

59.64l4

119.2828

86.5601

86.5601

198.80^6

166.0819

1G6.0819

298.2068
265.481*1
265.4841
417.4895
384.7668
381*. 7668
286.5989

286.5989

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3 2 +

10

3 2-

11

4 1 +

12

4 l -

13

4 2 +

14

4 2-

15

4 4 +

16

4 4 -

17

5 1 +

18

5 .7 -

19

5 2 +

20

5 2 -

para

16.2446

16.2446

56.0055

56.0055

45.0980

45.0980

115.6469

115.6469

10)1.7393

104.7393

195.1687

195.1687

l84.26.l2

184.2617

11*0.6310

140.6310

294.5708

294.5708

283.6631

283.6631

Table 3

i .

Basis set convergence of cross sections for collisions of
ortho-NH-, with He at a total energy of 100 . cm \ Basis designated
Bn includes the first n levels of ortho-Nil^ listed in table 2.

a( jke-O’k'e')) A c

jk £ k

-jJ'k’e'

E6

B9

B12

B15

0 0 +

0 0 +

156.

156.

156.

156.

10 +

1.22

1.18

1.18

1.18

2 0 +

0.'-l6

0.57

0.6l

0.67

3 3 +

0.067

O.092

o.n

0.12

3 3 -

0.91

0.90

0.94

0.93

10 +

10 +

162.

162.

162.

162.

20 +

0.99

0.99

1.02

1.03

3 3 +

0.95

0.95

0.93

0.94

3 3 -

o.o4i

0.045

0.047

0.049

2 0 +

2 0 +

158.

158.

158.

158.

3 3 +

0.27

0.31

0.33

0.36

3 3 -

0.75

0.77

0.80

0.81

3 3 +

3 3 +

85.5

85.2

84.8

85.1

3 3 -

o.hl

0.4i

0.43

0.43

3 3 - 3 3-

86.0

85.4 85.6

85.6

• • \\?

- I ■

ukl

Tuble 4

Basis set convergence of cross sections for collisions of

para-NH^ with He at a total energy of 100 cm . Basis designated

Bn includes the first n levels of para-NH^ listed in table 2.

o(jkc-»j , k , G'), f

ji_ k _£

J'fc'e 1

B6

BIO

Bl6

B20

11 +

11 +

l6l.

CN

M

•

l6l.

l6l.

11-

0.72

0.67

0.60

0.60

2 1 +

0.67

0.66

0.6l

0.6l

2 1-

0.16

0.16

0.16

0.17

2 2 +

0.0035

0.012

0.011

0.011

22-

1.07

1.01

1.03

1.03

21 +

2 1 +

163 .

162 .

161 .

l6l.

21-

0.49

0.46

0.38

0.37

2 2 +

0.81

0.74

0.74

0.74

22 -

0.11

0.13

0.12

0.12

2 2 +

2 2 +

165 .

165 .

165.

165 .

2 2 -

0.74

0.74

0.72

0.71

36

Tabic 5

Comparison of close coupling*

coupled.

states, and

effective

poten-

tial cross sections for

excitation

of ortho-NIIy

a( .Ike-* ,1'k'

€'), J ?

Energy, cm ^ J k c

J’k'e'

CC/B15

CS/B15

EP/B35

( corrected)

100. 00+

0 0 +

156.

196.

157.

157.

10 +

1.3.8

1.05

0.97

1.68

2 0 +

0.67

0.29

0.12

0.27

3 3 +

0.12

0.0

0.029

0.076

3 3 -

0.93

1.13

0.52

1.37

1 0 +

10 +

162.

163.

158.

158.

2 0 +

I.03

1.15

2.23

2.89

3 3 +

0.94

0.94

0.74

1.14

3 3 -

0.049

0.036

0.20

0.30

2 0 +

2 0 + ‘

158.

156.

146.

146.

3 3 +

0.36

O.29

1.04

1.24

3 3 -

0.81

0.8l

1.38

1.63

3 3 +

•3 3 +

89.

86.

77.

77.

3 3 -

0.43

0.45

0.43

0.43

3 3 -

3 3 -

86.

86.

75.

75.

190.

0 0 +

0 0 +

124.

125.

125.

125.

10 +

1.42

1.38

1.24

2.16

2 0 +

0.25

0.l4

0.24

0.54

3 0 +

0.45

0.46

0.30

0.80

3 3 +

0.0017

0.0

0.022

O.O58

3 3 -

2.65

2.71

. 1.63

4.32

4 3 +

0.0003

0.0

0.0025

0.0075

4 3 -

O.lG

0.17

0.l4

0.43

10 +

10 +

130.

130.

123.

123.

2 0 +

1.30

1 .30

2.90

3.75

30 +

0.12

0.15

0.21

0.32

3 3 +

2.24

2.30

2.07

3.16

3 3 -

0.10

0.11

0.47

0.71

4 3 +

0.10

0.11

0.15

0.27

4 3 -

0.35

0.33

0.20

0.36

37

Table 5> continued

Enerfry, cm ^ Jkc

J'kV

CC/B15

CS/B15

ET/B15

(corrected)

190. 2 0 +

2 0 +
3 0 +
3 3 +

3 3 -

4 3 +
4 3 -

143.

0.89

0.22

1.49

0.84

0.018

143.

0.90

0.24

1.56

0.84

0.028

134.

2.17

0.80

1.76

0.64

0.071

134.

2.57

0.95

2.08

0.86

0.095

3 0 +

3 0 +
3 3 +

3 3 -

4 3 +
4 3 -

16?.

0.77

0.27

0.051

0.97

162.

0.80

0.30

0.064

1.01

155.

1.10

0.47

0.24

0.82

155.

1.10

0.47

0.27

0.93

3 3 +

3 3 +

3 3 -

4 3 +

4 3 - •

153.

1.13

0.48

0.096

153.

1.19

0.49

0.11

153.

0.52

0.0073

0.081

153.

0.52

O.OO83

O.091

3 3 -

3 3 -

4 3 +
4 3 -

153.

0.11

0.47

153.

0.12

0.48

153.

0.03?

O.0095

153.

0.036

0.031

4 3 +

43 +
4 3 -

150.

0.38

l46. b
1.37 1 *

l*i9.

1.05

149.

1.05

4 3 -

4 3 -

150.

146. b

150.

150.

u Values corrected for "counting of states" obtained via Eq. (43) j
uncorrected values obtained via Eq. (44).

From a CS/B17 calculation.

Tt ?

HTTnoDuari’ or

Ottlfil nAL 1 t. . 1> m *

Table 6

Comparison of close coupling, coupler’, states, and effective poten-

cross sections

for excitation

of para -NII^ at a

total energy

of 250 cm

o(jke-J'k

v), A 2

J'kV

CC/B20

CS./JJ20

ET/B20

1

(corrected)

1 3 +

113 .

113 .

no.

no.

1 1 -

0.8l

O.89

0 . 4 l

0 . * 4 1

2 1 +

1.00

1.00

0.28

0.36

2 1 -

0.21

0.15

1.00

1 .29

2 2 +

o . 0006

0.0005

0.11

o.i 4

2 2 -

1.56

1.47

5.68

7.59

. 3 1 ■*

0.069

0 .l 4

0.052

0.080

3 l -

0.36

0.36

0.68

1 . 0*4

3 2 +

o. 4 i

0.42

1.20

1.83

3 2 -

0.22

0.49

0.69

1.05

*4 1 +

0.12

0.15

0.0J49

0.085

4 l -

0.0076

0.0050

o.:,2

0.21

4 2 +

0.053

O.053

0.13

0.22

4 2 -

0.033

0.048

0.11

0.19

4 4 +

0.044

0.0*45

0.015

0.026

4 4 -

1.24

1.15

0.94

1.63

2 1 +

122.

122.

U6.

ll6.

2 1-

0.62

0.65

1.22

1.22

2 2 +

0.87

0.89

2 . 5*4

2.54

2 2-

0.16

0.23

0.57

0.57

3 1 +

O.85

0.94

1.15

1.36

3 1 -

0.070

0.053

0.058

O.O 69

3 2 +

0.018

0.019

0.033

0.039

3 2 -

0.8l

0.85

1.6l

1.90

4 l +

0.019

0.055

0.026

0.035

4 1 -

0.085

0.089

0.16

0.21

4 2 +

0.20

0.20

0.38

0.51

4 2 -

0.075

0.13

o .]4

0.19

4 4 +

0.91

0 . 9*4

1.08

1.45

4 4 -

0.1*1

0.13

0.13

0.18

Tub]e 6,

continued

.Ike

.1'k’c'

cc/n?o

CS/B20

EP/B20

(corrected)

2 2 +

2 2 +

120.

119.

117.

117.

2 2-

1.33

1.35

0.69

O.69

3 1 +

0.15

0.15

0.24

0.28

3 1 -

0.29

0.28

O.69

0.82

3 2 +

0.91

0.82

o.i4

0.16

3 2 -

0.12

0.054

0.48

0.57

4 l +

0.038

O.O38

0.067

0.090

4 l -

0.039

0.070

0.13

0.17

4 2 +

0.015

0.13

0.022

0.030

4 2 -

0.21

0.22

0.22

0.30

4 4 +

0.068

0.066

0.048

0.064

4 l* -

0.0006

0.0007

0.0039

0.0052

3 1 +

3 1 +

l4l.

i4i. 1

134.

134.

3 1 -

0.30

0.34 l

0.76

0.76

3 2 +

0.77

0.82 ?

1.24

1.24

3 2 -

o.o4i

0.047?

0.16

0.16

4 l +

0.72

0.73 ?

0.97

1.10

4 l -

0.023

0.030°

0.043

0.049

4 2 +

0.018

0.022.

0.023

0.026

4 2 -

0.48

0.51 ?

0.56

0.63

4 4 +

0.17

0.19 l

0.24

0.27

4 4 -

0.51

0.52

0.68

0.77

l ' Valuer, corrected for "counting of ntates" obtained via Kq. (43);
uncorrected values obtained via Eq. (Mi).

b

From a CS/i3l6 calculation.